Problem: A particle moves along the curve $y=-x^3+5x$ so that that the $x$ -coordinate is increasing at a constant rate of $1$ unit per second. What is the magnitude (in units per second) of the particle's velocity vector when the particle is at the point $(-1,-4)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\sqrt{65}$ (Choice B) B $53$ (Choice C) C $4$ (Choice D) D $\sqrt{5}$
Background When working with motion along a curve, we should remember that this motion can also be represented by a position vector $(x,y)$. The main difference is that we're not given the equations for $x$ and $y$ in terms of time $t$, but only the relationship between $x$ and $y$ themselves. This however shouldn't prevent us from assuming the expressions for $x$ and $y$ in terms of $t$ exist. As always, $\vec{v}(t)=\left(\dfrac{dx}{dt},\dfrac{dy}{dt}\right)$. Setting up the math We are given that $\dfrac{dx}{dt}=1$ for any value of $t$. We are asked for the magnitude of the particle's velocity vector when the particle is at the point $(-1,-4)$. In other words, we are asked for $\left|\left|\left(\dfrac{dx}{dt},\dfrac{dy}{dt}\right)\right|\right|$ at the point $(-1,-4)$. Finding $\dfrac{dy}{dt}$ $\dfrac{dy}{dt}=-3x^2+5$ Finding $\dfrac{dy}{dt}$ at $(-1,-4)$ The expression for $\dfrac{dy}{dt}$ only depends on the particle's $x$ -coordinate, which in our case is ${x}={-1}$ : $\begin{aligned} \dfrac{dy}{dt}&=-3({-1})^2+5 \\\\ &=2 \end{aligned}$ Therefore, the particle's velocity vector at the point $(-1,-4)$ is $(1,2)$. Finding $||(1,2)||$ $||(1,2)||=\sqrt{5}$ In conclusion, the magnitude of the particle's velocity vector when the particle is at the point $(-1,-4)$ is $\sqrt{5}$ units per second.